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Feynman Diagrams and Low-Dimensional Topology

Maxim Kontsevich

We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, non-commutative geometry and several kinds of "topological physics."

The text below consists of 3 parts. The first two parts (topological sigma model and Chern-Simons theory) are formally independent and could be read separately. The third part describes the common algebraic background of both theories.

Conventions

Later on we shall use almost all the time the language of super linear algebra, i.e., the word vector space often means Z/2Z-graded vector space and the degree of a homogeneous vector v we denote by v.

In almost all formulas, one can replace C by any field of characteristic zero.

By graph we always mean finite 1-dimensional CW-complex. For g > 0 and n > 1 such that 2g + n > 2, we denote by Âig,n the

coarse moduli space of smooth complex algebraic curves of genus g with n unlabeled punctures.

1. Associative algebras and moduli spaces of algebraic curves: Two constructions

Let V be a differential associative algebra over C with an even scalar product on it. This means that V — VQ © V\ is a super vector space endowed with the structure of associative algebra such that

Vi • Vj C K+J-(mod 2),

with an odd derivation d, d2 = 0 and a scalar product ( , ) on each VQ, VI satisfying conditions

(1) (xy,z) = (x,yz),

(2) (x,dy) = (-lF(dxiy).

Denote by H(V) := Ker(d)/lm(d) the space of (co)homologies of the complex V. Then H(V) is again an associative super algebra endowed with the induced scalar product.

The first construction associates any differential algebra V with the scalar product (as above) such that

98 Maxim Kontsevich

1° dïmH(V) <oo , and

2° the induced scalar product on H(V) is not degenerate,

cohomology classes in Heven(M9in,C) for all g,n. In a sense the initial data for this construction is a kind of "non-commutative" homotopy type with Poincaré duality.

The second construction associates any differential algebra V with the scalar product such that

1° dim (V) < oo,

2° the scalar product on V is not degenerate,

3° H(V) = 0,

homology classes in HeYen(Mgìn, C) for all g,n. We will describe here the first construction. The idea arose on my read

ing [SI]. It goes through some generalization of the notion of differential algebra invented by Jim Stasheff [S2] many years ago.

Aoo-algebras

By definition, an Aoo-algebra (or, in other terms, strong homotopy associative algebra) is a collection (V, mi, 7712,...) where V is a super vector space and

mi : V —> V is an odd map, m2 : V (8> V —• V is an even map, m3 : V (8) V (g> V —> V is an odd map,

satisfying the higher associativity condition:

for any n > 1, x\,..., xn G VQ U V\

y ^ ±m^(xi 0X2 0 . . . ®xk-i®rrij(xk <g) . . . (8)xfc+<7-i) l < f c < i , l < j

0 Xfc+j 0 . . . 0 xn) = 0 ,

where the sign is given by the formula

_j_ _ c_]_y(xi+...+xfc_i)+0'-i)fe ^

Actually the associativity condition is an infinite sequence of bilinear equations on the multiplications 77 .̂

Feynman Diagrams and Low-Dimensional Topology 99

Examples:

n = 1: the corresponding equation is raiorai = 0. Hence mi is a differential and V is a complex.

n = 2: then ra2 : F<8)V —• V̂ is a morphism of complexes. V is a differential but not necessarily an associative algebra.

n = 3: the third equation means that rri2 is associative up to homotopy given by the map 7713.

We see that differential associative algebras are just Aoo-algebras with 777,3 = 7714 = . . . = 0. Conversely, for any ^^-algebra one can construct (applying the bar construction and then the cobar construction) a differential associative algebra which is in a sense homotopy equivalent to the initial Aoo-algebra. The advantage of ^4^-algebras is the possibility of transfer-ing AOQ-structures across quasi-isomorphisms of complexes ( "perturbation theory" in differential homological algebra, V.K.A.M. Gugenheim and J. Stasheff [GS]). In particular, one can construct a (non-unique!) structure of Aoo-algebra on H(V) which encodes all Massey operations arising on the space of cohomologies.

Aoo-algebras with scalar products

By definition, an ^l^-algebra with a scalar product is a /mfàe-dimensional ^oo-algebra V with a fixed nondegenerate even scalar product on V such that for any n > 1, the (n + l)-linear functional (ran, ) : Vr(^(n+1) _» V,

xi ® x2 ® . . . 0 xn.:-i 1—• {mn{xi <g>... 0 xn), xn+i)

is cyclically (i.e.Z/(n + 1)Z) symmetric (in the graded sense) for n odd and cyclically antisymmetric for n even.

We developed [K2] a perturbation theory for the case of algebras with scalar products and obtain higher multiplications on H(V) obeying cyclic-ity conditions as above for a non-commutative differential algebra V with Poincaré duality (i.e., H(V) is finite-dimensional and the induced scalar product on it is non-degenerate).

We shall construct for any ^^-algebra with a scalar product an even cohomology class on the moduli space of curves M9in. It is based on a certain combinatorial model for M9yn developed by J. Harer, D. Mumford and R. Penner, ([H], [P]).

Stratification of decorated moduli spaces

By definition, the decorated moduli space M^ is the moduli space of pairs (C, / ) where C is a compact connected complex algebraic curve of genus g and / : C —> R>o is a non-negative function which takes positive values

100 Maxim Kontsevich

exactly at n points. It is clear that the rational homotopy type of Mf*£ is the same as of MQin. The space Mf;% has a stratification with the strata equal to quotient spaces of euclidean spaces of some dimensions modulo actions of some finite groups.

Define a ribbon graph (or a fatgraph in other terms) as a graph with fixed cyclic orders on the sets of half-edges attached to each vertex. One can associate an oriented surface with boundary to each ribbon graph by replacing edges by thin oriented rectangles (ribbons) and glueing them together at all vertices according to the chosen cyclic order. A metric on the ribbon graph is a map from the set of edges to the set of positive real numbers R>o-

Denote by lZ9,n the moduli space of connected ribbon graphs with metric, such that the degrees of all vertices are greater than or equal to 3 and the corresponding surface has genus g and n boundary components.

Theorem 1.1. lZ9iU is isomorphic to M^e£.

This theorem follows from results of K. Strebel and/or R. Penner (see [Str], [PI] or an exposition in [H]).

It is clear that lZ9,n is stratified by combinatorial types of underlying ribbon graphs. This stratification was used before for the computation of the orbifold Euler characteristic of M9in ([HZ], [P]) and in the proof ([Kl]) of Witten's conjecture on intersection numbers of standard divisors on the Deligne-Mumford compactification Mg,n (see [Wl]).

The space 7Z9,n is a non-compact but smooth orbispace (orbifold), so there is a rational Poincaré duality between its cohomology groups and homology groups with closed supports with coefficients in the orientation sheaf. Hence one can compute the rational cohomology of M.9in using the complex generated as a vector space by equivalence classes of co-oriented strata.

State model on ribbon graphs

Let V be an Aoo-algebra with a scalar product and denote by v\,..., VN an orthogonal base of V (here N — dim (V)). We can encode all data in the sequence of cyclically (anti)-symmetric (in the graded sense) tensors with coefficients:

îi!,...,^ = K - i K ^ . - . O ^ . J , ^ ) , l<i*<7V n>2 .

Each ribbon graph defines a way to contract indices in the product of copies of these tensors. In other words, V defines a state model on ribbon graphs. The partition function Z(T) of a ribbon graph T is the sum over all


colorings of edges of T into N colors of the products over vertices of F of the corresponding coefficients of tensors T*. For example, the partition function of the skeleton of a tetrahedron is equal (with appropriate corrections of signs) to

/ v ^*1*2*5 ̂ 2*3*6^*1*4*3 ̂ 4*5*6 '

1 < Ù , . . . , Z 6 < A T

Actually the partition function is defined only up to a sign because, for odd n, the tensors T* are cyclically anti-invariant. One can check that the sign is fixed by a coorientation of the stratum corresponding to I\

Theorem 1.2. Y^Z(r)r is a well-defined cochain on M.^. It follows r

from the higher associativity conditions that this cochain is closed.

The proof of this theorem is a simple check. In a sense our construction of cohomology classes of M9in starting from an ^oc-algebra with a scalar product is analogous to famous constructions of knot invariants (see, for example, [RT1]). One can check that the resulting cohomology class does not depend on the choices in the construction and is a homotopy invariant of the differential associative algebra with a scalar product such that the induced scalar product on the cohomology space is non-degenerate.

There is a simple series of ̂ 4^-algebras with scalar products. The underlying vector space is an even one-dimensional space C1 . The scalar product is (1,1) = 1 and higher multiplications rrik are zero for odd k and arbitrary linear maps G®k —• C for even k. We have proved that the linear span of classes obtained from these algebras is the space of all polynomials in Morita-Miller-Mumford classes (the proof will appear elsewhere).

One could expect that it is possible to produce all classes of H*(A4gin) from the above construction. It would be interesting to construct examples of AOQ-algebras giving some new classes.

Also we expect that the structure of an Aoo-algebra with a scalar product appears naturally in the Floer homology of the space of free paths on almost any complex manifold and the corresponding cohomology classes are restrictions of the classes on Mg,n arising from the nonlinear sigma-model [Wl].

At the moment we don't know what kind of algebraic structure gives cohomology classes of the Deligne-Mumford compactification M9in. We hope that further development will lead to a better understanding of topological non-linear sigma-models, mirror symmetry and relations with matrix models of two-dimensional gravity.


Dual construction

We describe here a way to produce homology classes onjV(5)n. The starting ingredient is a finite-dimensional differential associative algebra V with a non-degenerate odd scalar product and trivial cohomology.

The right inverse to the scalar product can be considered as an odd element 6 of V 0 V. It follows from the compatibility of the scalar product with the differential that 6 is closed. Prom the triviality of H(V) it follows that there exists u; eV <S>V such that duo = 6. We can use a; as a "propagator" and tensors X ^ : V®k —> C

T(vi <g>...Vfc) = (viv2...vk-i,vk)

as "interactions." Again, we obtain a state model on ribbon graphs. Now we will consider the complex which is dual to the cochain complex from the previous section.

Theorem 1.3. y_\Z(X)T is a well-defined chain and it is closed. Its r

homology class does not depend on the choice of u.

The proof of this theorem is again a simple check. One can compute the pairing between cohomology classes arising from

Aoo-algebras and homology classes arising from differential associative algebras. The corresponding number will be the sum over all ribbon graphs of the product of two partition functions. One can identify this sum with the decomposition over Feynman diagrams of a finite-dimensional integral (see [BIZ] for an introduction to Feynman rules for mathematicians). Theoretically it gives a way to check non-triviality of classes arising from Aoo-algebras.

2. Perturbati ve Chern-Simons theory

Before the discussion of more complicated subjects we want to show the reader a simple formula for an invariant of knots in R 3 which arises naturally from perturbative Chern-Simons theory.

Denote by u(x) the closed 2-form ^ ^ijkx%dx\x\3dxk on R 3 \{0} ^ s t a n

dard volume element on S2 written in homogeneous coordinates). This form appears in the Gauss formula for the linking number of two noninter-secting oriented curves L\,L<i C R3:

#(LUL2) = J u>(x-y) x€Li,y€L2


Theorem 2.1. For a knot K : S 1 <-> R 3 where S1 = [0,1]\{0,1} the following sum

J u{K(h) - K{h)) A u(K{l2) - K{h)) 0<h<l2<h<U<l

+ j u(K(h)-x))Aw(K(l2)-x))Au{K(l3)-x)-±

0<Z1<Z2<Z3<l,xGR3\K(51)

is an invariant, i.e., does not change when we vary continuously K in the class of embeddings. This invariant is an integer number and it is equal to the second coefficient of the Conway polynomial of the knot K(S1).

This theorem follows from the study of certain compactifications of domains of integration and general properties of Vassiliev invariants (see below).

Overview of Chern—Simons theory

Let M be a closed oriented 3-dimensional manifold and let G be a compact Lie subgroup of the group of unitary matrices U(N). Denote by A a 1-form on M with values in the Lie algebra 0 of the group G. We can consider A as a connection in the trivial G-bundle over M.

The Chern-Simons functional of A is given by the formula

CS(A) = - i - / TV (AdA+%A*\ . 47r J M \ 3 /

This functional is invariant under infinitesimal gauge transformations 6A = dX + [A, X) for X : M -> 0 . The quantity ecs^ is invariant under all gauge transformations

A h—+ g-xAg + g-ldg, g : M -> G

and also could be defined for connections in nontrivial bundles. Formally we can write the integral over the quotient space of the space

of all connections in G-bundles over M modulo gauge transformations (for any integer k):

Zk = f(ecsM)kVA, k e Z .


Witten [W2] in 1988 described a way to "compute" the numbers Zk (for fc^O) starting from conformai field theory. These numbers are algebraic numbers belonging to cyclotomic fields. To compute Zfc, one has to decompose M into some simple pieces. Also one needs to know Ä-matrices for a quantum group with parameter q equal to a root of unity. Then the decomposition of M defines a way to contract indices in some tensor products of A-matrices and some other auxilary tensors. The result of the computation will be Zk- For a precise description see [RT2].

If one believes in the existence of Feynman integrals, then one expects a certain asymptotic behaviour of the complex numbers Zk for large fc. The integral must be concentrated near critical points of CS, i.e., near flat connections. Let us suppose for simplicity that there are only finitely many conjugacy classes of group homomorphisms n\M —» G. Conjecturally, there is an asymptotic formula of the following form:

Zk ~ £ ekCS^ - ^ - e x p ( f ) an,pk-A , k - +oo

where dp, R(p) can be computed from the Reidemeister torsion and the dimensions of the cohomologies of the adjoint local system ad(p), and an,p

are higher term corrections to the Gauss approximation. This expansion was checked (without higher corrections) for some

cases, where explicit formulas for Zk exists, see [G], [J], and for some more complicated cases using computer simulations in [FG].

Recipe for perturbation theory

Here we shall describe a way to define coefficients an?p in the conjectural asymptotic expansion of Zk. Let us fix a flat connection V in a G-bundle on M and denote by g the corresponding local system of Lie algebras (arising from the adjoint representation of G). Suppose for simplicity that g is infinitesimally irreducible and rigid,

i f ° ( M , g ) - i J 1 ( M , 0 ) = O .

It follows using Poincaré duality that g is acyclic. Connections near the flat connection V can be written as V + A where

A is a 1-form with values in g. The first step in the standard physical approach to the computation

of integrals over spaces of fields modulo gauge transformations is to add Faddeev-Popov ghost and anti-ghost fields. For mathematicians it means that we consider 0 and 2-forms with values in g as odd (=fermionic) fields.


The second step is to define a so-called gauge fixing. It is enough, for example, to choose a Riemannian metric on M. The gauge fixing gives a propagator which can be considered as a 2-form u) on M 2 with values in 0(g)0. This form is smooth and closed outside the diagonal Mdiag C M x M. The exterior derivative of UJ on all of M2 must be equal to the product of the delta-form on Mdiag times the inverse to the scalar product on 0. If M is a Riemannian manifold one can fix UJ uniqely as a harmonic form outside -Mdiag ( m other words, UJ will be a Green form).

The third and the last step is to compute integrals corresponding to Feynman diagrams. Let T be a finite nonempty connected graph with all vertices having degree 3 and without simple loops (that is, edges attached at both ends to one vertex). Such a graph has 2n vertices and 3n edges for some n > 1. Suppose that all vertices and all edges of V are numbered from 1 to 2n and to 3n respectively, and all edges are oriented. All these data can be encoded into two vectors /*,r* of length 3n, where k and Ti are labels of the left and the right vertex respectively attached to the edge i.

The integral corresponding to V is given by the formula

Ol , . . . , ;T2n)eM 2 n \ d i ag

where "Tr" denotes the tensor product over all vertices of skew-symmetric invariant.3-linear functionals on 0:

1 ^ 1 2 0 1 3 ^ Ti(X1X2X3 - X^X2Xl).

Here we use an isomorphism (g®2)®3n ~ (g®3)®2™ arising from the labeling of vertices and edges of I \

The n-th term in the perturbation expansion of Zk must be equal to

n->P (2n)!(3n)! ^

for some constant c ^ O . The factorial factors in the denominator come from counting of num-

berings of edges and vertices of F. There are two basic problems here:

(1) why the integrals converge, and

(2) why an^p does not depend on the choice of the form cu?


There are two slightly different solutions (1991) of these problems, by Axelrod-Singer [AS] and by Kontsevich [K2]. The first solution uses Green functions and Riemannian metrics, the second solution is softer in some sense but uses the triviality of the tangent bundle to M. We will describe here our approach.

Compactification of t h e configuration spaces

Let now M denote a compact manifold of arbitrary dimension. Denote by Confn(M) the space of configurations of n labeled distinct points in M,

Confn(M) = M n \a l l diagonals.

We shall construct a certain compact manifold with corners (i.e., a space looking locally like a neighbourhood of a point in the closed cube) Confn(M) containing Confn(M) as the interior.

The idea is to make all real-analytic blow-ups along all the diagonals

Mk c M n 5 i < fc < n _ i. For example, Cönf2(M) = Conf2(M) U SM is a manifold with the boundary 9Conf2(M) equal to the total space SM of the spherical bundle associated with the tangent bundle TM.

In general, Confn(M) is quite complicated. One can describe it as a set using the following notation:

Let V be a finite-dimensional vector space over R and m an integer greater than 1. Denote by Cm(V) the quotient space of Confm(y) modulo the action of the semi-direct product of the group V of translations and the group R+ of positive dilatations. The space Cm(V) is smooth because the action is free.

As a set, Confn(M) is equal to the disjoint union of some strata. Each stratum is a bundle over a configuration space Conf/~(M) for some /c, 1 < k < n. The fiber of this bundle over the point (x\,..., Xk) £ Conffc(M) is a product over some index set A of spaces Cmoc {TXia ) , a E A More precisely, strata correspond to abstract oriented forests with endpoints numbered from 1 to n and h connected components (trees). A is the set of vertices which are not endpoints, ia denotes the subtree containing a G A, ma is equal to the number of immediate successors of the vertex a.

One can check that the evident forgetful maps Confn(M) —• Confm(M) for n > m can be prolonged to smooth maps Confn(M) —• Confm(M). Another important fact is the list of codimension one strata in Confn(M). They are in one-to-one correspondence with subsets S c { l , 2 , . . . , n } with the cardinality # (5 ) > 2. Geometrically such a stratum corresponds to the situation when points with labels from S are coming close to each other.

An analogous construction in the context of algebraic geometry was invented by W. Fulton and R. MacPherson [FM]. It gives a resolution of


singularities of the union of all diagonals in the product of several copies of an algebraic variety.

Let us return to the problem with integrals appearing in the Chern-Simons perturbation theory. First of all, we fix a trivialization of the tangent bundle TM. It gives a decomposition

d (Conf2(Af)) =SM = S2 x M .

Define LJ on d (Conf2(M)) as the product of the standard volume element on S2 by the inverse of the scalar product in g. It follows from acyclicity of g that there exists a closed 2-form u globally on Conf2(M). Then the convergence of integrals is evident, because we integrate now a smooth form over a compact manifold with corners Conf2n(M).

Topological invariance of the sum of integrals follows from Stokes formula and the following general lemma:

Lemma 2.1. Let UJ G fi*"1^-"1) = O d _ 1 (C2(Rd)) denote any (antisymmetric volume element on 5 d _ 1 (for example, the standard rotation invariant volume element). For any integer N > 3 and for any two sequences k,ri, i = 1,. . . ,L of integers k ^ r^, 1 < U,Ti < N and for dimension d > 3 the integral

f L

(Xli...,xN)ecN(nd) l~l

has value zero.

The proof of this lemma is the following: The dimension of the integration space (Nd — d—1) must be equal to

the degree of the form (d — 1)L. It follows that in the graph associated with two vectors Z*,r* there exists a vertex with degree < 2. If the degree of this vertex is 0 or 1, then the form vanishes. If the degree of this vertex is equal 2, then fixing all vertices except this one we obtain the integral

/ u(x,y)tü{y,z)

y<ERd

considered as (d — 2)-form on the configuration space of 2 points x, z in Hd. This integral vanishes due to the involution

y «—> x + z - y


changing the sign. D

It follows from the lemma that only the simplest components of the boundary of Conf2n(M) give nonzero contribution to the variation of integrals, namely, they correspond to 2-element subsets in the set of indices. Then the Jacobi identity guarantees cancellation of all boundary terms.

The graph complex

Let us analyse the proof of topological invariance of an,p in the case of trivial p. Of course, this case is not realistic because for a nonempty manifold M or nonzero g, the cohomology group H°(M,g) is nontrivial. Nevertheless, imagine such a situation. It is clear that for any graph T the corresponding integral Zp is decomposed into the product of two factors,

ZY == Zr,(5 x ZY. M

where the first factor depends only on the Lie algebra (5 with nondegenerate scalar product and the second factor depends only on the manifold M.

The numbers Zr,<s, ^r,M depend also on some choices, namely, the choices of the cyclic order of 3-element sets of half-edges attached to each vertex. Both numbers change their sign when one changes the cyclic order at one vertex. We shall call an orientation of a 3-valent graph a choice of cyclic orders for all vertices up to an even number of changes. Thus ZY,&, ZY,M are both odd functional on oriented 3-graphs.

The only property of Zr,<s which we use in the proof is a kind of Jacobi identity:

ZYX,<ô + ZY2,® + ^r3,ö5 — 0

for triples of 3-graphs I \ obtained from a graph V with all vertices except one having degree 3 and the exceptional vertex having degree 4. Graphs I \ , i — 1, 2, 3 are obtained by "inclusion" of one edge in V instead of the exceptional vertex.

Let us define the graph complex as an abstract vector space over Q generated by equivalence classes of pairs (I\or) where T is a connected nonempty graph such that the degrees of all vertices are greater than or equal to 3 and (or) is an orientation of the real vector space R(edges of r ) 0 i7 1 ( r ,R) . We impose the relation

( I \ - o r ) = - ( r , o r ) .

It follows that (r , or) = 0 for every graph T containing a simple loop. The reason is that such graphs have automorphisms reversing the orientation in


our sense. One can check that for 3-valent graphs we have the same notion of orientation as above.

Define a differential d by the formula (for T without simple loops):

d(r,or) = Y^ ( r /e , induced orientation). e£{edges of T}

Here r / e denotes the result of contraction of the edge e, the "induced orientation" is the product of the natural orientation on the codimension-one co-oriented subspace R{edses of r ) / e c R{edses of r i and the orientation on ff1 ( r / e , R) ~ F 2 ( r , R ) . One can easily check that d2 = 0.

The differential d preserves the dimension of H1 (T) ( "number of loops" in physical language). Thus the graph complex is decomposed into the direct sum of its subcomplexes Cn}*, n > 1 consisting of graphs F with dim ( iJ1(r)) = n -f 1, the degree * being equal to the number of vertices of T. Each subcomplex Cn,* is finite-dimensional at each degree and has a finite length.

The homology of the graph complex is called graph homology. It is a challenging problem to compute it. We shall show now that graph homology consists of a kind of characteristic classes for diffeomorphism groups of odd-dimensional manifolds. It is clear that there are at least some nontriv-ial classes arising from Lie algebras with nondegenerate scalar products.

There exists a version of the graph complex which works for even-dimensional manifolds. One has to replace the vector space R{edges of r ) 0 Hx(r, R) by R<edges of r> in the definition of the orientation of a graph T.

Application of graph homology

Let M be compact oriented manifold of odd dimension d > 3 which has the rational homotopy type of a sphere, that is H*(M, Q) = H*(Sd

:Q). Remove one point p from M and consider the result as Rrf with the topology changed in a compact subset. Suppose that there exists a trivialization of the tangent bundle of M\{p} which coincides with the standard trivialization of THd near infinity.

Denote by £?Diff M a base of a universal smooth bundle with fibers diffeomorphic to M, with a section p and a trivialization of the tangent bundle to the fibers outside p having the same behaviour near p as above. We choose such notation because the homotopy type of this space differs from the classifying space of the diffeomorphism group Diff M by something quite simple.

Using the same technique as for Chern-Simons theory, we can construct differential forms on £?Diff M integrating products of copies of a suitably chosen form cu over the union of configuration spaces of fibers. These forms


are labeled by graphs with orientations in our sense. One can check that we obtain a morphism of complexes

cn,* -+ ndn-* (mmM\ .

Thus graph homology maps to the cohomology of JBDiff M. In the case d — 3 any top-degree class in H*(Cn) gives a zero-degree

cohomology class of £?Diff M, i.e., just a real number, which is an invariant of M. For example, the simplest non-zero class represented by a graph with 2 vertices and 3 edges connecting both vertices gives an invariant of a homology 3-sphere with spin structure. The formula for this invariant is

/ u;3

C2(M\{p})

Theorem 2.2. This invariant is also invariant under homology trivial spin cobordisms.

This theorem was proven by Cliff Taubes [T].

Problems:

(1) find a geometric construction of these invariants,

(2) prove (or disprove) that graph homology maps to cohomology classes with rational coefficients,

(3) generalize all this to the case of nonhomology spheres with nontrivial tangent bundles and try to incorporate also the group algebra of the fundamental group.

Knots invariants and Vassiliev's theory

Let us use the same notations for Chern-Simons theory as before. For a simple closed oriented curve L in a 3-manifold M and for a finite-dimensional unitary representation p of the group G, denote by WP,G the gauge-invariant functional (Wilson loop) on the space of connections given by the formula

WPìG{A) — trace in p of the monodromy of the connection A along L .

Then for any family of loops Li and representations p^, i = 1 , . . . ,n and


for any k one can write the Feynman integral

Zk((L1,Pl),...,(Ln,Pn))= ff[WPi,Li (ecs^)kVA.

Witten [W2] gave rules to compute this integral assuming that the curves Li do not intersect and are also framed (the normal bundles to L* are trivialized). One can write down terms in the perturbation series for these integrals as well.

Let us restrict ourselves to the case M — S3. Witten showed that these integrals are just versions of Jones invariants of links (see also [RT1]). Because TTIS3 = 1 there is only one critical point for the Chern-Simons functional, the trivial connection. Thus, integrals corresponding to Feynman diagrams will be the product of algebraic and geometric factors.

We can replace S3 by the non-compact space R 3 and use as the propagator the Gauss form

, N 1 (xi-yi)d(x^ -yi)Ad(xk-yk) u{x,y) = —tijk ; [ö •

Sn J \x-y\ò

The simplest invariant of knots obtained in this way gives a formula which we presented just before the review of Chern-Simons theory (cf. Theorem 2.1). It is hard to say who wrote this formula first, probably D. Bar-Natan or E. Guadagnini, M. Martellini and M. Mintchev [GMM]. It is quite strange that this simple formula (just a generalization of the Gauss formula for the linking number) was invented so late (1988-1989), and not by topologists, but by physicists. Until 1991 (S. Axelrod-I.M. Singer and my work) it was not clear whether higher order integrals converge or not.

It was recognized by D. Bar-Natan and myself that perturbative Chern-Simons theory for knots is closely related with Vassiliev invariants (see [VI], [V2]).

In 1988-1989 V. Vassiliev introduces a class of knot invariants using an idea absolutely independent from Witten 's. His approach was the following:

Let us consider the space of embeddings as the complement in the infinite-dimensional vector space of all maps from S1 to R 3 to the closed subspace of maps with the self-intersecting or singular image. We intersect both spaces — the space of knots and its complement, with an appropriate generic family of finite-dimensional vector spaces with increasing dimensions. For example, the spaces of trigonometric polynomial maps of fixed degrees will do. Then it is possible to apply the usual Alexander duality.


Of course, we can generalize this to the case of embeddings of an arbitrary manifold into Euclidean space of arbitrary dimension.

The main technical invention of V. Vassiliev is a very simple simplicial resolution of singularities of the space of non-embeddings, which allows one to compute its homology groups with closed support. This technology can be applied to a very broad class of situations, and in good cases it gives a complete description of the weak homotopy type of some function spaces. The case of knots turns out to be borderline. The spectral sequence arising in Vassiliev's approach does not converge well. The zero-degree part of its limit is a certain countable-dimensional subspace in the continuum-dimensional space of all cohomology classes. In particular, the space of knot invariants considered as the 0-degree cohomology group of the space of embeddings contains a countable-dimensional subspace of Vassiliev invariants.

We now give their definition. For any knot K : S1 °-> R 3 and any family of nonintersecting balls

£?i, £?2, • . . , Bn C R

such that the intersection of any ball with K(S1) looks from above like one line passing over another, one can construct 2n knots. These knots will be labeled by sequences of +1 and —1 of length n. The knot K€lì.„ì€n obtained from K — K(S1) by replacing for all i such that a = —1 the part of the knot in the interior of Bi by another standard sample with the first line passing under the second line. Of course, K+ i ? + iv . .5 + i is the initial knot K.

Let $ be a knot invariant with values in an abelian group A (for example, A = Z,Q, C,. .).

Definition. <£ is an invariant of degree less than n if for all K and # i , . . . , Bn as above the following equality holds:

] T 61 . . -6n^>(K e i , . . . ,€J=0

For k — 0 , 1 , . . . denote by Vk the vector space of Q-valued invariants of degree less than k + 1. The space of all Vassiliev invariants V — I ) Vk is

k the space of invariants of finite degree. There is an evident generalization of the notion of Vassiliev invariants to the case of knots and links in arbitrary 3-dimensional manifolds.

We want to mention here several features of these invariants (see [BN], [K4], [VI], [V2]):


(1) All terms in the perturbati ve Chern-Simons theory are invariants of finite order.

(2) The space of invariants of a fixed degree is finite-dimensional, there exists an a priori upper bound on its dimension. Moreover, this space is algorithmically computable. Unfortunately, the only known method to compute this space for a fixed degree takes super-exponential time.

(3) For any Vassiliev's invariant, there exists a polynomial-time algorithm for computing this invariant for arbitrary knots.

(4) It is not hard to prove that if all Vassiliev's invariants for two knots coincide, then their (Alexander, Conway, Jones, Kauffman, HOMFLY, etc.)-polynomial invariants coincide.

(5) The class of Vassiliev invariants coincides with the class of invariants arising from the work of V. Drinfel'd [D] on quasi-Hopf algebras.

One can show that every Vassiliev invariant has a representation as a finite-dimensional integral over some configuration space associated with the pair of manifolds K(S1) C R 3 of a product of copies of the Gauss form. We proposed in [K4] also another integral formula for Vassiliev invariants based on [D]. Conjecturally both integral formulas give the same answer.

Vassiliev invariants could be considered as top-degree homology groups of a complex closely related with the graph complex. Exploiting Vassiliev's ideas and perturbative integrals we proved the following fact:

Theorem 2.3. For n > 4 let Xn be the space of embeddings of S1 into Rn. There exist finite-dimensional vector spaces over Q,

V^,VÌ'j, 0<j<i

such that for any n > 4, k > 0

(n-3)i+j=k

The proof is organized as follows:

(1) the second term of the Vassiliev spectral sequence gives an estimate above for cohomology groups of Xn,

(2) Feynman diagrams give differential forms on Xn, so produce a lot of cohomology classes,

(3) using combinatorial arguments we show that cohomology of the complex of Feynman diagrams coincide with the second term of the Vas-


siliev spectral sequence. Thus estimates above and below coincide and the spectral sequence collapses here.

The proof shows also that @^ V/'° is equal to the space V of Vassiliev knot invariants.

Complete exposition of this theorem will appear elsewhere.

3. (Non)commutative symplectic geometry

Here we will show relations between parts 1 and 2 which appear to be very diffferent.

Three infinite-dimensional Lie algebras

Let us define three Lie algebras. The first one, denoted by Zn, is a certain Lie subalgebra of derivations of the free Lie algebra generated by 2n elements p i , . . . ,p n , # i , . . . , qn. By definition, ln consists of the derivations acting trivially on the element YUPi^i}-

The second Lie algebra an is defined as the Lie algebra of derivations D of the free associative algebra without unit generated by p i , . . . , p n , c i , . . . , qn satisfying the condition D {^2(piqi - qiPi)) = 0.

The third Lie algebra cn is the Lie algebra of polynomials F G Q[pi , . . . ,Pn,gi , . . . ,</n] such that F(0) = F'(0) = 0, with respect to the usual Poisson bracket

{F,G}=j:(dFdG dFdG

One can define cn also as the Lie algebra of derivations of a free polynomial algebra Q[p*, q*\ preserving the form ]T dpi A dqi and the codimension one ideal (pi , . . . ,pn ,Çi,--- ,Çn)-

Our aim is computation of the stable homology (with trivial coefficients) of these Lie algebras. The spirit of the (quite simple) computations is somewhere between Gelfand-Fuks computations (see [F]) and cyclic homology.

If we denote by hn one of these three series of algebras, then we have a sequence of natural embeddings hi C hi C . . . C h^ where the last algebra corresponds to the case of a countable infinite number of generators. Of course, H^hoo) =™ H*(hn).

For the limit algebras h^ we have a structure of Hopf algebra on its homology (as is usual in K-theory). The multiplication comes from the homomorphism fooo 0 h^ —• h^ (oo + oo = oo in set theory) and the comultiplication is dual to the multiplication in cohomology.


This Hopf algebra is commutative and cocommutative. Thus if*(ftoo) is a free polynomial algebra (in the Z/2Z-graded sense) generated by the subspace PH*(hoc) of primitive elements.

In all three cases we have an evident subalgebra sp(2n) C hn consisting of linear derivations. The primitive homology of sp(2oo) is well-known:

P f 7 ( / 9 ì n ì f Q & = 3(mod4) PHk(sp(2oo)M) = [0 M 3 ( m o d 4 )

Now we can state our main result:

Theorem 3.1. For all three cases PH^hoo) is equal to the direct sum of PHk(sp(oo)) and

(1) (for the case l^)

0 #2n-2- fc(OutFree(n), Q) , n>2

where OutFree(n) denotes the group of outer automorphisms of a free group with n generators,

(2) (for the case a^)

n>0,2-2g-n<0

(3) (for the case Coo)

0 (graph homology) £n) . n>2

The gradings on the homology groups arising from the natural grading on hoc are equal to (2n — 2), (4p — 4 + 2m) and (2n — 2), respectively.

Idea of the proof of Theorem 3.1

Recall that our Lie algebras hn are Z>0-graded. Thus the standard chain complex f\*{hn) is graded. We consider the case when n is much larger than the grading degree.

It is well known that every Lie algebra acts (through the adjoint representation) trivially on its homology. The algebra sp{2n) C hn acts reduc-tively on f\*(hn). Hence the chain complex is canonically quasi-isomorphic to the subcomplex of sp(2n)-invariants. It is easy to see that this subcom-plex stabilizes when n —* oo.

We claim that for the case c^ the subcomplex of invariants is a version of the graph complex constructed from all not necessarily connected graphs with degrees of all vertices > 2.


The underlying vector space of the Lie algebra cn as a representation of sp(2n) is equal to

where V — Q(pi , . . . ,p n , # i , . . . , qn) is the defining 2n-dimensional representation of sp(2n).

Thus our chain complex as a representation of sp(2n) is equal to the sum

0 (Ak*(S2(V)®Ak*{S3{V)®...) . fc2>0,fc3>0,...

Every summand is a space of tensors on V satisfying some symmetry conditions.

From the main theorem of invariant theory, it follows that invariant tensors are obtained by contraction of indices. Then one can easily identify all possible ways to contract indices with appropiate graphs.

Consider the subcomplex consisting of all non-empty connected graphs. It contains as a direct summand the complex of graphs with degrees of all vertices equal 2. This part gives the primitive stable homology of sp(2oo). The rest is quasi-isomorphic to our standard graph complex using some spectral sequence arguments. This proves Theorem 3.1 for the commutative case.

In the associative and the Lie case we need some explicit description of the corresponding Lie algebras:

Theorem 3.2. Let V denote a standard symplectic vector space Q 2 n .

(1) an is isomorphic to (&k>2{V<S>k)Z^kZ as sp(2n)-module,

(2) ln is isomorphic to (&k>2{V®k <S> Lk)^k where Lk is the (k - 2)!-

dimensional representation of the symmetric group T,k with character

X(lfc) = (fc-2)!, x ( l V ) = (ò-l)!o6-V(a), x(ab) = - (6-l)!a6-V(a)

and %(*) = 0 for all other conjugacy classes of permutations (\i is the Möbius function).

For the proof of this theorem we developed a language of non-commutative geometry. In a sense an and ln are Lie algebras of hamiltonian vector fields in non-commutative associative geometry and Lie geometry respectively (see [K3]).

In the associative case, the main theorem of invariant theory gives a version of the graph complex consisting of ribbon graphs. Thus we can


use stratification of the moduli space of curves (Theorem 1.1) and obtain Theorem 3.2 for a^.

Denote by G^ for n > 2 the set of equivalence classes of pairs (r, metric) where T is a nonempty connected graph with Euler characteristic equal (1 — n) and degrees of all vertices greater than or equal to 3, (metric) is a map from the set of edges to the set of positive real numbers R>o- One can introduce a topology on G^ using the Hausdorff distance between metrized spaces associated in the evident way with pairs (r, metric). It is better to consider G^ not as an ordinary space, but as an orbispace (i.e., don't forget automorphism groups). We mention here that G^ is a non-compact and non-smooth locally polyhedral space. It has a finite stratification by combinatorial types of graphs with strata equal to some quotient spaces of Euclidean spaces modulo actions of finite groups.

A fundamental fact about the topology of G^ is the following theorem of M. Culler and K. Vogtmann (see [CV]):

Theorem 3.3. G^ is a classifying space of the group OutFree(n) of outer automorphisms of a free group with n generators.

We can construct a finite cell-complex homotopy equivalent to BOutFree(n) passing from the natural stratification of Q^ to its barycen-tric subdivision. The corresponding cochain complex carries a filtration by graphs (by the minimal graph corresponding to the strata attached to the cell). Computations show that the spectral sequence associated with this filtration collapses at the second term to the Lie version of the graph complex. D

Graph homology also has an interpretation in terms of the moduli space of graphs:

Hk(Cn,,) = H$£1(g(n\e),

where e is one-dimensional local system given by orientations of graphs.

Strong homotopy algebras and their characteristic classes

A structure of A^-algebra (=strong homotopy associative algebra) on a super vector space V is equivalent to an odd derivation d of the free associative super algebra without unit generated by F* (g> C0 '1 satisfying the equation [d, d] = 0. Also an Aoc-algebra with a scalar product is the same as a homomorphism of Lie superalgebras


Here ak\i denotes the Lie super algebra constructed from the symplectic super vector space Q2fcl* in the same way as an is constructed from Q 2 n

(see Theorem 3.2). One can define in an analogous way a strong homotopy Lie algebra with

a scalar product as a homomorphism C0 '1 —• c* ®C and a strong homotopy associative commutative algebra with a scalar product as a homomorphism C0 '1 —> Z* 0 C. Mention here that the definition for strong homotopy Lie algebras uses the commutative version of hamiltonian vector fields and vice versa. In many senses the associative case is self-dual.

The homomorphism C0 '1 —> h* 0 C composed with the inclusion h* -̂> ^oo|oo gives a map

C 1 = H2k{CQ\1) - H2k(h00\00) ® C = H2k(hOQ) ® C

for all positive integer k. Thus any strong homotopy algebra with a scalar product produces

classes in the corresponding version of the graph complex. As in the associative case, there are two constructions starting from differential algebras with scalar products giving characteristic classes.

Perturbative Chern-Simons theory near the trivial representation can be reformulated as the pairing between characteristic class arising from a finite-dimensional Lie algebra with a scalar product (considered as a differential algebra with d — 0) and the class arising from a differential commutative algebra(=the de Rham complex of a 3-manofold) with a non-degenerate odd scalar product and trivial cohomology.

Topological applications

There are two obvious functors

{commutative algebras} —> {associative algebras} —> {Lie algebras} .

They give morphisms of the corresponding Lie algebras and maps between their homology groups.

Thus we obtain maps

H*(Mg,n) -> H*(BDiffM,R)

for any g,n > 1 and for any 3-dimensional rational homology sphere M. In particular, at degree zero, we obtain a number which is an invariant associated with the topological type of an oriented surface with boundary and of the 3-manifold M. Also one can increase the dimension of M and make some shift in degrees.


Witten [W3] proposed an "explanation" of this map. Let us choose a generic Riemannian metric on M. Then the tangent space TM can be endowed with an almost complex structure. Moreover, it is formally a 3-dimensional almost Calabi-Yau manifold. It follows from the index theorem that (the virtual) dimension of the space of holomorphic curves in TM with boundary at the zero section is equal zero. The "number" of such curves of genus g with n boundary components must be the same number which we obtained using graphs. Also, if one considers a cycle in M)iff M then one obtains a cycle in A49in-

Unfortunately, it seems at the moment that this construction will not work well and must be improved.

Let us now consider the Lie algebra l^. It is clear that it is a Lie analogue of the Teichmüller group T9ii for large genus g —• oo. One can make this analogy more precise and construct (using Theorem 3.1) maps

#fc(OutFree(n), Q) - i J ^ ^ ^ o o , Q ) .

We don't know what is the geometric or "physical" meaning of these maps.

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Max-Planck-Institut für Mathematik Gottfried-Claren-Str. 26 D-53225 Bonn, Germany

Received February 1, 1993 Revised March 28, 1994

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